where all the particles are assumed to be outgoing. The soft-factor is usually written in terms of the soft-photon polarization vector for which a particular choice has been made in this formula. Charge conservation is then derived from the fact [1]} that the Lorentz invariance of the \(S\) -matrix also requires it to be invariant under on-shell gauge transformation of the soft-photon polarization vector, i.e, \(\epsilon _\mu (q) \rightarrow \epsilon _\mu (q) + \alpha q_\mu \) , where \(q_\mu \)  is the 4-momentum of the soft-photon. Now, for the purpose of deriving global charge conservation, instead of the on-shell gauge-invariance one can use the boundary condition (REF ). This is also more natural in our context. Of course, on-shell gauge invariance and the falloff condition in (REF ) are both consequences of Lorentz invariance of the \(S\) -matrix. We would like to note that this is similar to the situation in 2-D CFT where the fall-off condition \(T(z)\sim 1/z^4\)  gives the Ward identities for the global conformal group starting from the correlation function of \(T(z)\)  and a set of primaries.
